423.
ON THE PRESSURE DEVELOPED IN A LIQUID DURING THE COLLAPSE OF A SPHERICAL CAVITY.
[Philosophical Magazine, Vol. xxxiv. pp. 94—98, 1917.]
WHEN reading 0. Reynold's description of the sounds emitted-by water in a kettle as it comes to the boil, and their explanation as due to the partial or complete collapse of bubbles as they rise through cooler water, I proposed to myself a further consideration of the problem thus presented; but I had not gone far when I learned from Sir C. Parsons that he also was interested in the same question in connexion with cavitation behind screw-propellers, and that at his instigation Mr S. Cook, on the basis of an investigation by Besant, had calculated the pressure developed when the collapse is suddenly arrested by impact against a .rigid concentric obstacle. During the collapse the fluid is regarded as incompressible.
In the present note I have given a simpler derivation of Besant's results, and have extended the calculation to find the pressure in the interior of the fluid during the collapse. It appears* that before the cavity is closed these pressures may rise very high in the fluid near the inner boundary.
As formulated by Besant*, the problem is—
"An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant."
Since the fluid is incompressible, the whole motion is determined by that of the inner boundary. If U be the velocity and R the radius of the boundary at time t, and u the simultaneous velocity at any distance r (greater than R} from the centre, then
u/U=R*/r*- .................................(1)
* Besant's Hydrostatics and Hydrodynamics, 1859, § 158. |r |2 = tanh2& ................ (64)